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Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right)=\left|\begin{array}{ccc}{\left(1+\mathrm{x}\right)}^{\mathrm{a}}& {\left(1+2\mathrm{x}\right)}^{\mathrm{b}}& 1\\ 1& {\left(1+\mathrm{x}\right)}^{\mathrm{a}}& {\left(1+2\mathrm{x}\right)}^{\mathrm{b}}\\ {\left(1+2\mathrm{x}\right)}^{\mathrm{b}}& 1& {\left(1+\mathrm{x}\right)}^{\mathrm{a}}\end{array}\right| \\ \mathbf{Constant}\mathbf{ }\mathbf{term}\mathbf{ }\mathbf{will}\mathbf{ }\mathbf{be}\mathbf{ }\mathbf{obtained}\mathbf{ }\mathbf{by}\mathbf{ }\mathbf{putting}\mathbf{ }\mathbf{x}\mathbf{=}\mathbf{0} \\ \Rightarrow \mathrm{f}\left(0\right)=\left|\begin{array}{ccc}{\left(1+0\right)}^{\mathrm{a}}& {\left(1+0\right)}^{\mathrm{b}}& 1\\ 1& {\left(1+0\right)}^{\mathrm{a}}& {\left(1+0\right)}^{\mathrm{b}}\\ {\left(1+0\right)}^{\mathrm{b}}& 1& {\left(1+0\right)}^{\mathrm{a}}\end{array}\right| \\ \Rightarrow \mathrm{f}\left(0\right)=\left|\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\\ 1& 1& 1\end{array}\right| \\ \mathrm{As} \mathrm{all} \mathrm{three} \mathrm{rows} \mathrm{are} \mathrm{same} \\ \Rightarrow \mathrm{f}\left(0\right)=0 \\ \mathrm{Hence} \mathrm{options}\left(\mathrm{a},\mathrm{c},\mathrm{d}\right) \mathrm{are} \mathrm{incorrect}. \\ \mathbf{\Rightarrow }\mathbf{Option}\left(\mathbf{b}\right)\mathbf{ }\mathbf{is}\mathbf{ }\mathbf{correct}\mathbf{.} \end{gathered}$$

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